What is the speed of the rocket relative to the Earth?

AI Thread Summary
The discussion revolves around calculating the speed of a rocket fired from plane B towards plane A, with both planes moving at relativistic speeds relative to Earth. The user initially calculates the rocket's speed as 0.217c but believes it should be negative due to the direction of motion. However, it is clarified that the rocket moves in the same direction as plane B from the Earth's perspective, leading to the conclusion that the rocket's speed is indeed -0.56c. The user expresses doubt about the book's solution, suggesting possible errors in the provided information. The conversation highlights the complexities of relativistic velocity addition and the importance of frame of reference in such calculations.
terryds
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Homework Statement



Plane A flies with speed 0.6c chasing plane B which speed is 0.4c . Both speed is measured by observer on Earth. Then, plane B fires a small rocket which rest mass is 10 kg towards plane A. Rocket speed is 0.2c relative to plane B where c equals the speed of light in vacuum.
What's the speed of the rocket relative to the Earth?

Homework Equations



##V = \frac{V' + u}{1 + \frac{V' u}{c^2}} ## (reverse Lorentz transformation)

The Attempt at a Solution



So, I think the stationary frame is the Earth. The moving frame is plane B. The event is the rocket.

I put
V' = -0.2 c (because A chases B, then B fires a rocket towards A which means opposite direction of the plane), u = 0.4 c (because the moving frame is plane B, I define positive direction is the direction of the plane)

But, I get V = 0.217 c which means that the rocket has the same direction to those planes according to the observer in the Earth.
I think it should be negative sign.
Please help me where I got wrong.

The solution is -0.56 c but I don't know how to figure it out
 
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terryds said:
Please help me where I got wrong.
Easy. You are not wrong. Well, except for:
terryds said:
I think it should be negative sign.
From the Earth frame, the small rocket will move in the same direction as B, since the relative speed between B and the rocket is smaller than the relative speed between B and the Earth frame.
 
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Orodruin said:
Easy. You are not wrong. Well, except for:

From the Earth frame, the small rocket will move in the same direction as B, since the relative speed between B and the rocket is smaller than the relative speed between B and the Earth frame.

So, the solution is wrong, right? I also doubt the book since it's just written by my seniors hahaha.. thank you very much
 
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